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The interplay of optical force and ray-optic behavior between Luneburg lenses Alireza Akbarzadeh, J. A. Crosse, Mohammad Danesh, Cheng-Wei Qiu, Aaron J. Danner, and Costas M. Soukoulis ACS Photonics, Just Accepted Manuscript • DOI: 10.1021/acsphotonics.5b00352 • Publication Date (Web): 24 Aug 2015 Downloaded from http://pubs.acs.org on September 3, 2015

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The interplay of optical force and ray-optic behavior between Luneburg lenses Alireza Akbarzadeh1,*, J. A. Crosse2, Mohammad Danesh3,2, Cheng-Wei Qiu2, Aaron J. Danner2, and Costas. M. Soukoulis1,4 1

Institute of Electronic Structure and Laser, Foundation for Research & Technology-Hellas, Heraklion,

Crete, Greece 71110 2

Department of Electrical and Computer Engineering, National University of Singapore, 4 Engineering

Drive 3, Singapore 117576 3

Electronics and Photonics Department, Institute of High Performance Computing, 1 Fusionopolis Way,

Singapore 138632 4

Ames Laboratory and Department of Physics and Astronomy, Iowa State University, Ames, Iowa, USA

50011 *

Corresponding author: [email protected]

Abstract. The method of force tracing is employed to examine the optomechanical interaction between two and four Luneburg lenses. Using a simplified analytical model as well as a realistic numerical model, the dynamics of elastic and fully inelastic collisions between the lenses under the illumination of collimated beams are studied. It is shown that elastic collisions cause a pair of Luneburg lenses to exhibit oscillatory and translational motion simultaneously. The combination of these two forms of motion can be used to optomechanically manipulate small particles. Additionally, it is addressed how fully inelastic collisions of four Luneburg lenses can help us achieve full transparency as well as isolating space to trap particles. Keywords: Optical force, Geometrical optics, Optical Manipulation, Graded-index media, Metamaterials

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The mechanical properties of light has been the topic of debate from the early stages of the emergence of electromagnetic wave theory [1-5]. The optomechanical properties of light have been the subject of much study, but despite this the area of optical force is still an active area of research and specifically the enigma of the momentum of light in a medium is still under scrutiny [6-11]. The well-known Abraham-Minkowski dilemma, the definition of the Poynting vector, and the interpretation of the stress tensor as well as its relation to the optical force and torque densities are the major concerns of the current fundamental studies on the optical force. In spite of these theoretical uncertainties, optical forces have been employed practically in fascinating applications; microscopy and optical imaging [12, 13], optical tweezers and particle trapping [14-16], optical tractor beams and optical lift [17-22], light driven motors [23], and radiation pressure on optical cloaks [24, 25] are few examples out of many. With the advent of new fabricating and characterization technology and development of theoretical studies, it is expected that optical force will play a big role in novel applications in the near future. The conventional approach to calculate the optical force in a medium is to solve the Maxwell equations for the electromagnetic fields, construct the elements of the stress tensor and integrate the divergence of the stress tensor over the volume of the interest [26, 27]. However, applying this standard full-wave method in complex media can be technically hard or intensively time-consuming. Alternatively, by taking advantage of the ease provided by the geometrical optics and relying on its sufficient accuracy under some restrictions, which are met in many scenarios, the authors proposed a new method called “force tracing” to trace the optical force field along the trajectories of light in a complex medium [28]. In [28], the authors took the

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8/17/2015 electromagnetic fields as quasi-plane waves with slowly changing amplitudes and rapidly fluctuating phases as,

r r r  E ( rr , t ) = E0 exp ik0 k ⋅ rr − iωt  r r r r r  H ( r , t ) = H 0 exp ik0 k ⋅ r − iωt

(

)

(

(1)

)

r where ω is the angular frequency, k is the wave vector, k0 = ω c , c is the speed of light in free r r space, and both E0 and H 0 are vectors with approximately constant magnitudes. On the basis of this assumption, which is the core of geometrical optics, and with the help of the Hamiltonianbased ray equations [29, 30], after a lengthy algebraic manipulation the Lorentz force density in a lossless isotropic medium is simplified to

r f

normalized

=

1 r r k × Lk n4

(2)

r where n is the refractive index and Lk = eˆz  k x ( dk y dτ ) − k y ( dk x dτ )  . It should be mentioned r that the force density in equation (2) has been normalized by ε 0 E0

2

2 with ε 0 being the free

space permittivity. An interested reader is referred to [28] for more details on the derivation of equation (2), as well as the proposed formulation for the force tracing in anisotropic media and the surface force density at the interfaces between two different media. However, it should be noted that the anisotropic formulation in [28] is true only for the cases with diagonal constitutive tensors and for non-diagonal cases it should be modified. In this article, we use force tracing technique to study the dynamics of optomechanical interaction between Luneburg lenses under illumination of collimated light beams. As will be discussed later, owing to the spatial variation of optical forces acting on a system made up of more than one Luneburg lens, the collision between the lenses causes oscillatory and

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8/17/2015 translational motions. Both elastic and fully inelastic collisions will be considered and through a simplified analytical as well as a more realistic numerical model, the equation of motion will be derived and discussed in detail. The collision of Luneburg lenses under illumination of light leads to optical transparency, space isolation and spatially-temporally modulated light beams. These results are interesting for a wide range of researchers and can have important applications in biosensing, particle imaging, particle transport and particle trapping.

Discussion and Results A Luneburg lens [29] is a spherically symmetric graded-index lens with profile index

n ( r ) = 2 − r 2 which focuses parallel rays shining from one side to a single point (Figure 1(a)). Using the force tracing technique, it can be seen that the parallel rays entering the lens would exert a positive force (a force in the direction of the light ray) on it (Figure 1(b)). However, as shown in Figure 1(c), if rays enter into the lens from a single point, they actually apply a negative force onto it, and this is due to the fact that the horizontal components of the momentum of the rays gradually increase after departing the lens. If we have an even number of Luneburg lenses touching each other sequentially, the total change in the momentum of light entering parallel to a line cutting the touching points would be zero and the combination of Luneburg lenses do not feel any force. In other words, from the geometrical optics point of view two Luneburg lenses are complementary media and a point source on one of the lenses is perfectly imaged on the other lens (Figure 1(e)). As an example (see Figure 1(d)), if we have two Luneburg lenses, after travelling through this combination the exiting rays look as if there existed no object in front of them (aside from 180o rotation) and, hence we achieve transparency with inversion. However, if the lenses detach from each other, not all the rays that impinge on

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8/17/2015 the first lens would reach the second one and consequently full transparency would be absent. In addition, the momentum of light would alter during its journey through the separated lenses and Luneburg lenses have some optically mediated interaction with each other. By studying the force on each lens, as shown in Figure 2, we see that as all the incoming rays arrive at the first lens, the force on the first lens is always constant and acts in the direction of the wave vector of light. However, the optical force on the second lens is a function of the distance from the first one and can be attractive, repulsive or vanishing. Figure 3 shows the total force acting on each lens as a function of the separation, as calculated via a full-wave simulation in COMSOL. It can be seen that the force on the second lens is attractive for small separations and hence the second lens can become trapped close to the first lens. As seen in Figure 3, the force acting on the second lens is a complicated function of position. In order to obtain a quantitative understanding of the motion of the lenses, let us assume that the force on the first lens is constant, F1 ( x1 ) = F , and the force on the second lens is a linear function of the separation, F2 = − F + α ( ∆x − R ) , where the separation is given by ∆x = x2 − x1 with R = R1 + R2 the minimum separation owing to the finite radius of the lenses. Here, α is a scaling factor and should be chosen with care in order to make the linear model resemble a realistic scenario. This form of the force is analytically tractable but preserves the basic phenomenology of the system and is a reasonable approximation for small separations where the force on the second lens remains negative. The approximated linear force for α = 4.65 ×10−12 is shown in Figure 3, as well. In the following we consider two lenses of identical mass m1 = m2 = m and radius R1 = R2 = R 2 , with initial positions and velocities, x1,0 , x2,0 and u1,0 and u2,0 for the two lenses, respectively.

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8/17/2015 The constant, positive force F1 causes the first lens to undergo linear motion. However, the force acting on the second lens F2 is a negative restoring force which leads to the collision between the two lenses and hence an oscillatory motion. It proves to be convenient to change variables from the absolute coordinates of the spheres to the center-of-mass coordinates and the separation. In this coordinate system the linear and oscillatory motions separate and hence can be solved individually. The equation of motion for the separation ∆x , reads (see Figure 4) d 2 ∆x d 2 x2 d 2 x1 F2 − F1 α ( ∆x − R ) − 2 F = − 2 = = dt 2 dt 2 dt m m

(3)

 α   α  d ∆x α ( ∆x0 − R ) − 2 F = sinh  t  + ∆u0 cosh  t dt αm  m   m 

(4)

 α   α  2F m α ( ∆x0 − R ) − 2 F  cosh  t  + ∆u0 sinh  t + +R α α  m   m  α

(5)

which solves to

∆v =

and

∆x =

1

where ∆x0 = x2,0 − x1,0 and ∆u0 = u1,0 − u2,0 are the initial separation and initial approach velocity, respectively. Let us consider the case where the lenses start at rest u1,0 = u2,0 = 0 (hence, ∆u0 = 0 ). The

first collision occurs when the separation ∆x = R , i.e. when

 α  2F α ( ∆x0 − R ) − 2 F  cosh  t + =0 m α α   1

This leads to a first collision time of

6


(6)

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t1 =

m

arccosh [ B0 ]

(7)

2F 2 F − α ( ∆x0 − R )

(8)

α

where,

B0 =

In an elastic collision both the energy and momentum must be conserved. From momentum conservation we have u1,1 + u2,1 = v1,0 + v2,0

(9)

2 2 2 2 u1,1 + u2,1 = v1,0 + v2,0

(10)

u1,1 = v2,0

(11)

u2,1 = v1,0

(12)

and from energy conservation

which leads to

where v1,0 and v2,0 are the incoming velocity of the lenses before the collision and u1,1 and u2,1 are the outgoing velocity after the collision. Thus, the outgoing velocity from the first collision is

∆u1 = −∆v0 =

2F B0 α m

B02 − 1

(13)

The outgoing velocities ∆u0 → ∆u1 = −∆v0 and new initial position ∆x0 → ∆x1 = R gives the initial conditions for the second collision. All the subsequent collisions can be computed recursively. The ith collision occurs at

 α   α  αm 1 − cosh  ti  + ∆ui −1 sinh  ti  = 0 m 2 F m     which gives a collision time of

7


(14)

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ti =

m

α

arccosh [ Bi −1 ] + ti −1

(15)

where

Bi −1 =

4 F 2 + α m ∆ui2−1 4 F 2 − α m ∆ui2−1

for i > 2

(16)

and a separation velocity of ∆ui =

2F αm

Bi2−1 − 1 − Bi −1∆ui −1

for i > 2

(17)

As can be seen from this analytical formulation, the two lenses, in the absence of damping, undergo infinite number of elastic collisions as long as the light is shining. However, if the effect of damping is present, which is the case in realistic situations, the model needs to be modified. In order to take the damping into account, we introduce a damping factor kd in the differential equation (3),

d 2 ∆x d ∆x F2 − F1 α ( ∆x − R ) − 2 F + kd = = 2 dt dt m m

(18)

Equation (18) solves to

∆x =  D1 cosh ( β t ) + D2 sinh ( β t )  e− kd t 2 +

2F

α

+R

(19)

where, β = 4α + kd2 m 4m , D1 = α ( ∆x0 − R ) − 2 F  α and D2 = ( 2∆u0 + kd D1 ) 2β . From equation (19) it is seen that under the influence of damping the oscillatory motion of the system decays. Consequently, it is predicted that as the collisions go on, the range of fluctuations become smaller and smaller; ultimately the lenses touch each other and the negative restoring force between the lenses vanishes. The oscillations vanish and lenses remain in contact.

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8/17/2015 The center-of-mass xc , can be rewritten as xc = ( x2 + x1 ) 2 = x1 + ∆x 2 . As the first lens experiences a constant force, the equation of motion for x1 is easily solved. Hence the center-ofmass motion can be written as xc ( t ) =

F 2 t + ∑ ( u1,i −1t + x1,i −1 + ∆xi ( t ) 2 )  χ ( t − ti −1 ) − χ ( t − ti )  2m i

(20)

where χ ( t ) is the Heaviside step function, i.e. χ ( t ) = 1 for t ≥ 0 and χ ( t ) = 0 for t < 0 , and ∆ xi is computed via the recursive formula above. Thus we see that the center-of-mass undergoes

parabolic acceleration with a correction owing to the interaction between the lenses. As matter of fact, the whole system under the illumination of light exhibits two types of motion; the local oscillatory motion of the two lenses and the global translational motion of the whole system. However, in the realistic scenario the force applied on the second lens is a complicated function of position. As analytical solution for this complicated force is not possible, we invoked a numerical analysis to obtain the equation of motion of the two lenses. The result of such analysis is shown in Figure 5(a) and 5(b). In Figure 5(a), where the positions of the two lenses with respect to time for six collisions are presented, the translational and oscillatory motion of the lenses can be clearly observed− a fact expected from the simplified analytical formulation. Shown in Figure 5(b) is the distance between the spheres versus time, which is a cyclical shaped curve confirming the oscillatory motion of the spheres. Note that the force acting on the first lens is constant and it may seem that its motion should be a parabolic function with respect to time. However, the velocity of the first lens changes abruptly at the moments of collision, which inflicts discrete corrections on the parabolic equation. Hence, as seen in Figure 5(a), the path of the first lens x1 ( t )

looks like an oscillatory function superimposed on a parabola. The

approximated linear model for small initial separations should produce results which are

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8/17/2015 reasonably close to those of the realistic model. For the purpose of accuracy validation, for the case ∆x0 = 0.1 our results corresponding to the realistic and analytical (with the scaling factor

α = 4.65 ×10−12 ) models are presented in Figure 6(a) and 6(b). Note that the linearized F2 ( x ) for α = 4.65 ×10−12 is shown in Figure 3, where it is seen that the linear and realistic graphs for F2 ( x ) match each other quite well for small separations. Comparing the graphs provided in Figure 6, we see that the analytical model is in a good agreement with the numerical model, which certifies the validity of the employed approximation. We believe that the combination of the oscillatory motion and the translational motion of the two lenses can be utilized in applications such as imaging, trapping or time-space-modulated transportation of small particles. The addition of damping and loss causes the amplitude of the oscillations decay along the translational motion. This is shown in Figure 7(a) and 7(b), where the path of the lenses and distance between them are depicted versus time. It is seen in Figure 7 that the period of collisions and the velocity of the lenses decrease as time passes and the lenses approach each other more and more. Eventually the oscillations are damped away and the net force on the whole system of lenses tends to zero, which makes the damping effect dominate and both the translational and oscillatory motions stop. The dynamic analysis presented above is for the completely elastic collision between the Luneburg lenses. However, if the lenses with velocities v1 and v2 collide in a completely inelastic way, then after the collision the lenses carry on their movement together at velocity

( v1 + v2 )

2 . This inelastic collision in principle can be attained by employing mechanisms which

cause the lenses stick together after the first collision imposing gradually increasing friction on the system or introducing a proper potential barrier. The immediate consequence of such an

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8/17/2015 inelastic collision is the full transparency that we can achieve by combining two pairs of Luneburg lenses, as depicted in Figure 8(a). As seen in Figure 8(a), due to the complementarity in each pair of Luneburg lenses, the well collimated rays enter the lenses with no reflection, gradually travel through lenses and exit as if there were no object in front of them. Shown in Figure 8(b) is another setup which can be acquired by the inelastic collision of Luneburg lenses. As noticed in Figure 8(b), four Luneburg lenses can be attached together by illuminating collimated beams in horizontal and vertical directions. In this way, the shared space between the lenses is almost empty of light rays and, due to the symmetrical structure of the setup and light rays, the optical force acting on particles within this space is very small or vanishing. This space can be used to trap, sense or transport particles. In conclusion, we reviewed the method of force tracing briefly and being inspired by this method we considered the optical force applied on a system consisting an even number of Luneburg lenses. We studied the dynamics of the collision occurring between the lenses under the illumination of collimated light beams analytically and numerically. We showed that for the elastic collision the system exhibits translational and oscillatory motion, while for the fully inelastic case it has only translational motion. Finally, we discussed how effectively such a lightmatter interaction can be used in various applications.

Acknowledgments Alireza Akbarzadeh and Cheng-Wei Qiu gratefully appreciate the initial fruitful discussions with Professor Juan José Sáenz. Work at FORTH was supported by the European Research Council under the ERC Advanced Grant No. 320081 (PHOTOMETA). Work at Ames Laboratory was

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8/17/2015 partially supported by the US Department of Energy (Basic Energy Science, Division of Materials Science and Engineering) under Contract No. DE-AC02-07CH11358.

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(a)

(b)

(d)

(c)

(e)

Figure 1. (a) Ray trajectories in a Luneburg lens; (b) a collimated beam shining from left-side pushes the Luneburg lens; (c) rays entering from a single point pull the Luneburg lens; (d) the performance of two touching Luneburg lenses; (e) the complementarity of two touching Luneburg lenses. A point located on one of the two Luneburg lenses is imaged perfectly on the other one.

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Figure 2. The optical force field applied by the light rays illuminating from left side onto two non-touching Luneburg lenses. The optical force on the first Luneburg lens is always pushing and constant. The optical force on the second lens can be pushing, pulling or null subject to its distance from the first lens.

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Figure 3. The total optical force applied on the two Luneburg lenses calculated via the full-wave simulation and the approximated linear optical force acting on the second lens for α = 4.65 × 10 −12 . It is assumed that the summation of the Luneburg lenses’ radii is equal to R . When ∆x − R = 0 , the lenses are touching.

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Figure 4. The schematic of the two Luneburg lenses which are colliding due to a collimated light beam illuminated from left side.

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(a)

(b)

Figure 5. (a) The paths of the two spheres obtained from the realistic numerical model for six collisions. At the collision moments the two curves touch each other. (b) The corresponding distance between the two spheres versus time for the six collisions. ( m = 10 −10 , R1 = R2 = 1 , x1,0 = 0 and x2,0 − R = 0.5 )

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(a)

(b)

Figure 6. (a) The paths of the two spheres obtained from the simplified analytical model and the realistic model for five collisions. At the collision moments the two curves touch each other. (b) The corresponding distance between the two spheres versus time for the five collisions, where the solid-line graph is representing realistic model and the dashed-line curve representing the analytical model. ( m = 10 −10 , R1 = R2 = 1 , F1 = 9.4 × 10−13 , x1,0 = 0 ,

x2,0 − R = 0.1 and α = 4.65 × 10 −12 )

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(a)

(b)

Figure 7. (a) The paths of the two spheres obtained from the realistic numerical model under the influence of damping. (b) The corresponding distance between the two spheres versus time. ( m = 10 −10 , R1 = R2 = 1 , x1,0 = 0 ,

x2,0 − R = 0.5 and kd = 0.03 )

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(a)

(b)

Figure 8. (a) The full transparency achieved by inelastic collision of two pairs of Luneburg lenses; (b) inelastic collision of four Luneburg lenses under vertical and horizontal shining of the collimated light beams to make an isolated area in between.

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For Table of Contents Use Only The interplay of optical force and ray-optic behavior between Luneburg lenses Alireza Akbarzadeh, J. A. Crosse, Mohammad Danesh, Cheng-Wei Qiu, Aaron J. Danner, and Costas. M. Soukoulis

In this paper we consider two and four Luneburg lenses under illumination of collimated light beams. With the use of the method of force-tracing we show how the Luneburg lenses can exhibit translational and oscillatory motions. We also show that the presence of damping causes the two lenses gradually stick together and stop. Therefore we can obtain the full transparency. We also discuss how the opto-mechanical interactions of the Luneburg lenses can be used in different applications like particle trapping, space isolation and particle transport.

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Interplay of Optical Force and Ray-Optic Behavior ... - ECE@NUS

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